Umakant Kondapure, Collin Fernandes, Nipun Bhatia, Vikram Bathula and, Ketki Deshpande Solutions for Chapter: Rotational Motion, Exercise 3: Competitive Thinking

From a uniform circular disc of radius $R$ and mass $9 \mathrm{M},$ a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is:

A cord is wound around the circumference of wheel of radius $r$. The axis of the wheel is horizontal and moment of inertia about it is $I$. The weight $mg$ is attached to the end of the cord and falls from rest. After falling through a distance $h$, the angular velocity of the wheel will be

A mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

Two solid cylinders $P$ and $Q$ of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder $P$ has most of the mass concentrated near its surface, while $Q$ has most of its mass concentrated near the axis. Which statement
$\left(s\right)$ is/(are) correct?

A body at rest starts from top of a smooth inclined plane and requires $4 \sec$ to reach bottom. How much time does it take, starting from rest at top, to cover one-fourth of a distance?

A hollow sphere of mass $M$ and radius $R$ is rotating with angular frequency $\omega$. It suddenly stops rotating and $75\%$ of kinetic energy is converted to heat. If $\mathrm{S}$ is the specific heat of the material in $\mathrm{J} {\left(\mathrm{kg} \mathrm{K}\right)}^{-1}$ then rise in temperature of the sphere is ($\mathrm{M}.\mathrm{I}.$ of hollow sphere $=\frac{2}{3} {\mathrm{MR}}^2$)

An automobile moves on a road with a speed of $54 \mathrm{km} {\mathrm{h}}^{-1}$. The radius of its wheels is $0.45 \mathrm{m}$ and the moment of inertia of the wheel about its axis of rotation is $3 \mathrm{kg} {\mathrm{m}}^2$. If the vehicle is brought to rest in $15 \mathrm{s}$, the magnitude of average torque transmitted by its brakes to the wheel is

A ceiling fan rotates about its own axis with some angular velocity. When the fan is switched off, the angular velocity becomes ${\left(\frac{1}{4}\right)}^{\mathrm{th}}$ of the original in time $t$ and $n$ revolutions are made in that time. The number of revolutions made by the fan during the time interval between switch off and rest are (Angular retardation is uniform)